To solve the problem (REF ), we
transform the variational inequality () into a variational equality. For this
purpose, we apply an active-set algorithm, as described in [1]}, [2]}, [3]}, which allows to identify
the subdivision into stick/slip/open regions for every \(\Gamma _i\) . At a given step of the active-set algorithm, the stick/slip/open regions of each fracture \(\Gamma _i\)  are fixed and the inequality () becomes:
\(\mathcal {R}_{t,i}= \left( \mu , g \right)_{\Gamma _{i}^{\text{stick}} }+ \left( \mu _N, g_N\right)_{\Gamma _{i}^{\text{slip}}}+ \frac{1}{k} \left( \mu _T, t_{T}^h - t_T^*\right)_{\Gamma _{i}^{\text{slip}}}+ \frac{1}{k} \left( \mu , t^h\right)_{\Gamma _{i}^{\text{open}}}= 0, \qquad i=1,\ldots ,n_f,\) 
